Parallax is the apparent shift in position
of a nearby object when it is viewed from
different locations.
Those might be two different observatories
on the Earth's surface
(as was the case in the famous
transits of Venus
in the eighteenth and nineteenth centuries),
but,
in most astronomical applications,
are two spots in Earth's orbit around the Sun:

If one knows the value of the baseline distance b,
and can measure the parallax angle π

Yes, I know this is confusing. Yes, astronomers back in the
old days should have picked a better letter.
Sometimes you see the "curly pi" used to denote this angle,
but few font families include it

then one can use simple geometry to compute the
distance to the target object d.

Q: Write a simple formula for d.

Of course, that simple formula ignores many complications.
For example,

the target object is usually located far above or below the
plane of the Earth's orbit

the target star and the Sun are both moving through
the Milky Way Galaxy

The importance of parallax lies in
its fundamental simplicity:
we humans understand geometry, and trust it.
As long as we can make measurements of the angular displacements
over time,
we can use the resulting distances with confidence.

There may be minor issues involving corrections for the
small shifts of the background reference stars themselves;
but if we choose to use galaxies or quasars
as the reference sources, we can avoid those completely.

Most important of all,
parallax is a direct method of measuring distances.
It relies on a single assumed quantity:
the astronomical unit (AU) = the distance from Earth to Sun.
Back in the 1800s and early 1900s,
this was indeed a major issue,
and astronomers
great efforts
to pin down the value of the AU.
But these days, thanks to radar and spacecraft flying
through the solar system,
we do have a very, very good idea for the size
of the AU.
For example,
the JPL Horizons system
uses a value taken from
the Planetary and Lunar Ephemerides DE430 and DE431,
which list

1 AU = 149597870.700 km

That's ... quite a few signficant figures.

And that's it.
Heliocentric parallax
does NOT require any assumptions about
stellar luminosity or evolution,
or the distribution of sizes and colors of stars,
or the spectral energy distribution of
glowing hydrogen gas.

So, if astronomers can measure a distance via parallax,
they will rely on it much more than distances
determined by other methods.

So, parallax is a good method.
But how far out into space can it reach?

Back in the old days, when parallax was performed by optical
telescopes from the surface of the Earth,
it was very difficult to achieve precisions of better
than about 0.01 arcsec.
Let's use milliarcseconds (mas) from this point forward,
as it will be more convenient.
So, good ground-based measurements had precisions
of perhaps 10 mas.

However, no such devices could make all-sky surveys.
As a result, catalogs of stellar distances were incomplete,
as well as being rather low in precision.
For example,
the
Bright Star Catalog ,
compiled over the period of 1964 to 1991,
listed parallax values for
about 2700 stars.

Q: What is the most common value for parallax (mas)?
Q: If we could measure stars perfectly, what should
the most common value be?

The fact that this histogram turns over below
15 mas is a sign that the precision
of the typical measurement is ... about 15 mas.

Q: If the uncertainty of a typical measurement is
15 mas, how far away can a star be
and still have a "good" distance?

It turns out that the nature of parallax measurements
means that the connection between the error in the measurement
and the error in the distance
is not simple.
There are a number of tricky aspects which can make the
analysis of, say, the possible bias in some catalog,
a complicated matter.

The first big problem is that the quantity of interest,
the distance to a star d,
is INVERSELY related to the measured angle π.

This means that an error of (for example) 20 percent in the measurement
does NOT mean that there will be an error of 20 percent in the
computed distance.

Why not? Well, let's do an example to find out.
Suppose that we measure the parallax to a star to be

If the uncertainty is the usual "1-sigma" variety,
then this measurement means that the true value
of the parallax angle
has a 66% chance of lying in the range 80 to 120 mas.

Fine so far. But what does that mean for distance
we derive from this angle?
We need to calculate the distance corresponding
to the angle (100 - 20) and the angle (100 + 20).
I'll let you fill in the table below.

The fundamental issue here is that a realistic,
symmetric error in the measured angle
leads to an ASYMMETRIC error in the derived distance.
The larger the fractional error in parallax angle becomes,
the more asymmetric this error in the derived distance.
For example, if we measure

then the range of distances consistent with the
measurement at 1-sigma looks like this:

We can sum up this first problem:
symmetric errors in a measured parallax angle
lead to asymmetric errors in the derived distance ...
with a larger range of possible true distances
on the "farther away" side.

The really nasty bit of the errors-in-parallax discussion
is a consequence of the first conclusion:
symmetric errors in angle lead to asymmetric errors in distance.
Let's go back to this example again:

Based on the parallax angle π = 10 mas,
we derive a distance of d = 100 pc ---
but due to the uncertainty in the angle,
the true location of the star has a 66% chance
of lying anywhere within the range of
59 - 333 pc.

What that means is that the TRUE location of the star could be here ...

or here

or here

In fact, the star could lie anywhere along a line in this direction,
with equal probability, based on that one measurement.
In a one-dimensional universe, the probability that the TRUE
location of the star is farther than 100 pc is larger than
the probability that the star is closer than 100 pc,
simply by the ratio of the lengths of the possible locations:

Our course, our real universe is THREE-DIMENSIONAL.
Sorry, I can't draw a good 3-D figure showing the
two regions, but I hope you can extrapolate from
the two-dimensional case to the three-dimensional one.

Q: What is the ratio of probabilities now, in 3-D?
probability that star is farther than 100 pc
--------------------------------------------- =
probability that star is closer than 100 pc

Now, what's the big deal about all of this?
The big deal is that
we live in a region of relatively uniform stellar density.
In our neighborhood of the Milky Way, there are
roughly equal numbers of stars in all directions:
left, right, up, down, forward, back.
Yes, there tend to be more stars in the plane of the
Milky Way, but the thickness of the disk is (until Gaia)
larger than typical distance we could measure via
parallax, that doesn't really matter.

And so, suppose that we measure the parallax to a star
to have the value 10 +/- 1 mas,
corresponding to a measured distance of 100 pc.
But there are three possibilities for the TRUE distance
to this star:

the star could actually be closer to the Earth,
but a negative error in the measured parallax
(taking a true value of 11 mas to measured value of 10 mas)
makes it appear to be farther away

the star could actually be at exactly 100 pc

the star could actually be farther from the Earth,
but a positive error in the measurement
(taking a true value of 9 mas to a measured value of 10 mas)
makes it appear to be closer

Which of these is most likely?
The answer is "C".
Because there are more stars in the "more distant" volume,
it is more likely that a distant star has been
improperly measured than a nearby star.
The result is that our measurements of parallax
lead to a systematic bias: stars appear closer
than they really are
(sort of like dinosaurs.)

The size of this bias depends (mostly) on the ratio
of the uncertainty in parallax to the value of parallax:

Let's look at some examples.
We will observe a set of stars with
measured parallax angles of exactly 10.0 mas,
which means measured distances of exactly 100 pc.

When the fractional error is very small -- just a few percent --
then the errors in the derived distances,
and the overall bias,
are pretty small.

But as the fractional uncertainty grows,
the errors and bias increase.
Even uncertainties of just 10% can cause big mistakes
in the measured parallax.
Notice the long tail of true distances out to 140 pc
and beyond;
the errors in those derived distances are 40 percent or more,
far larger than the 10% uncertainty in the parallax angle itself.

If the uncertainty becomes 20% or larger,
the results involve errors which are frequently
a factor of 2 or more.
No one should try to use such measurements for
any careful calculations.

Things are just silly when the uncertaity is 30%!

This table summarizes the results of a simulation
in which I created a region of space with uniform
stellar density, "measured" enough stars to find
1000 falling within a particular range of parallax
values around 10 mas, and then compared those
"measured" distances to the actual distances.

The moral of this story is -- don't trust distances
based on parallax unless the uncertainty in the measurement
is much smaller than the measurement itself.
I would prefer not to rely on any measurements
in which the fractional uncertainty is larger than 10 percent.

Phew. Now that we understand the weaknesses of the
parallax method, we can try to answer this question.
If we adopt as a general rule that most
ground-based measurements of parallax had uncertainties
of about σπ = 10 mas,
then we can place a very rough limit on the
distances of stars derived from such measurements:

Gosh, that's not very far.

Of course, astronomers DID use parallax measurements which were
considerably smaller than 100 mas;
scientists are always interested in pushing the limits
of their data.
But this is a warning that any statistical inferences
based on ground-based measurements of stars at greater
distances must be very carefully considered,
and corrections must be made for the Lutz-Kelker effect.

The Hipparcos satellite was launched in 1989 and
its first catalog was released in 1997.
This project immediately improved our knowledge of
stellar distances (and motions, and luminosities)
by a large amount.
Not only did it cover the entire sky,
and measure stars fainter than those typically
found in parallax catalogs
(down to about mag 10),
it also had a typical precision of
about
2 mas.
In other words, its typical measurement was about as good
as the very best ground-based measurements.

Q: If the uncertainty of a typical measurement is
2 mas, how far away can a star be
and still have a "good" distance?

One of the wonderful results of Hippacos' measurements
were
the very accurate HR diagrams
which could finally
be made,
since we now had LOTS of distance measurements to stars
of all types.
Below is one made using Hipparcos stars with
uncertainties of ≤ 10% in their parallax angles.

Gaia is a rather unusual "telescope,"
since it has been specially designed for astrometric observations.
It resembles the Hipparcos satellite in many ways.
The basic structure is built around two triple-mirror telescopes
which point at regions in the sky separated by
an angle of about 106.5 degrees.

Light from the two telescopes is combined so that it falls
upon the focal plane simultaneously.
The telescope rotates with a period of six hours,
causing stars to sweep across this focal
plane in about 60 seconds.
Objects in both directions are detected and measured
together.

Figure 1 from
Lindegren et al. 2016. Note that this figure
is rotated by 180 degrees relative to previous schematic.
The original caption follows.

Layout of the CCDs in Gaia's focal plane. Star images move
from left to right in the diagram. As the images enter the field of view,
they are detected by the sky mapper (SM) CCDs and astrometrically
observed by the 62 CCDs in the astrometric field (AF). Basic-angle
variations are interferometrically measured using the basic angle monitor
(BAM) CCD in row 1 (bottom row in figure). The BAM CCD in row 2
is available for redundancy. Other CCDs are used for the red and blue
photometers (BP, RP), radial velocity spectrometer (RVS), and wave-
front sensors (WFS). The orientation of the field angles η
(along-scan, AL) and ζ (across-scan, AC) is shown at bottom right.
The actual origin (η, ζ) = (0, 0) is indicated by the
numbered yellow circles 1 (for the preceding field of view)
and 2 (for the following field of view).

The telescope slowly precesses, so that its spin sweeps out
new areas in the sky over time.
A typical star will be measured about 70 times
over the course of the primary mission (2014 - 2019);
that's about 14 measurements per year, though not all equally spaced.

As the spacecraft precesses, the "partners" of each star will gradually
change.
Eventually,
the spacecraft will have many millions (billions?) of measurements
of the relative positions of millions of stars.
Scientists can then use a honking-big linear algebra procedure
to solve simultaneously for the positions and motions of each star.

Q: What is the CURRENT standard uncertainty in the
parallax for a bright star in DR1?
Look in Table 1 or Figure 7 of Lindegren et al. 2016
Q: Using that uncertainty, how far out into space
will Gaia make good distance measurements?

As time passes, and Gaia continues to scan the skies,
it will build up larger and larger sets of measurements
of each star, allowing it to create a better
solution for the position, parallax, and proper motion
of each object.
A recent paper by de Bruijne, Rygl and Antoja
provides some predictions
for the FINAL Gaia precisions.

Q: What is the prediction for the FINAL standard uncertainty
in the parallax of a bright star, when Gaia
finishes its survey?
(Look in the figure above)
Q: Using that uncertainty, how far out into space
will Gaia make good distance meausurements?

In any case, Gaia will greatly increase our knowledge of
the stellar neighborhood.
It will produce a catalog of roughly 100 million stars
with distances good to about 10 percent.
The Hipparcos catalog contains only 0.12 million entries,
of which only about 0.04 million satisfy the
Lutz-Kelker criterion.

As we will see in a future lecture,
the star
RR Lyrae
is a very important one.
It is one of the nearest and brightest of a class
of pulsating stars which can help us
to measure distances not only within the Milky Way,
but to other galaxies.

Clearly, if we wish to use the class of RR Lyr stars
as distance indicators, we need to know very precisely
how luminous they are ... and, so, how far away they are.
Let's consider the star RR Lyr itself.

This is, of course, just one star. But if our estimates of
the distances to other stars, and other galaxies,
depend upon RR Lyr stars in particular,
and if the new Gaia data suggests that our old value might
be incorrect by such an amount ...
well, that should make you start to worry about
the upper rungs on the Cosmological Disetance Ladder!

Based on the
earlier discussion of errors in parallax ,
we might expect that the Hipparcos measurements,
with their larger uncertainties,
would show a systematic bias relative to the more
precise Gaia values.
Let's put that to the test.

I went to the
Gaia DR1 archive site
and chose the "ADQL Search Form",
which allows the user to enter SQL-like queries to
a number of catalogs.

The result was a set of about 93,000 stars measured by both
Hipparcos and Gaia.
I then compared the parallax values on a star-by-star
basis.
As the graph below shows, Gaia and Hipparcos
do agree very well, on average:
the red dots are individual stars,
and the blue symbols are medians within bins of width 1 milliarcsec.

If we zoom in, however, we can see some asymmetry:
the Gaia parallax angles tend to be SMALLER
than the Hipparcos angles,
and the difference grows as the angles decrease in size.

In other words,
since the Hipparcos angles are slight OVER-estimates
(due to the Lutz-Kelker bias),
the Hipparcos distance values tend to be UNDER-estimated
as on approaches the limit of its measurements.

We can see that clearly if we plot DISTANCES on our
graph instead of angles.

Of course, optical telescopes are not the only ones which
can make parallax measurements.
Any telescope on (or around) the Earth will
share the annual journal,
and so have a chance to measure the parallactic shifts
of celestial sources.

Stars emit most of their energy in the optical region
of the spectrum, and very little in the radio;
it isn't possible to detect most stars with radio telescopes.
Clouds of gas and dust do emit plenty of radio waves,
but they are for the most part large, extended sources,
not the compact objects one requires for parallax measurements.

However, there are some circumstances which do allow radio
astronomers to apply the power of interferometry
to make very precise measurements of the positions of
compact radio sources, and therefore to determine the
distances to those sources via parallax.
The key element is the maser emission from
small clumps of gas within some star-forming regions.
Astronomical masers arise within compact volumes,
thus appearing as point sources to our radio telescopes,
and produce radiation of very pure frequency.
The combination allows astronomers to determine their positions
with very small uncertainties.

One example of parallax measurements in the radio regime
is provided by the
Japanese VERA project.
VERA is a array of four radio dishes stretched out across
the Japanese archipelago.

If one looks at star-forming regions, such as S269,
with an optical telescope, one sees clumps of stars
immersed in gas and dust.
If one looks with a radio interferometer, one can
see much finer details.
Compare the scales on the IR picture at left
with the VERA map at lower right.

If one measures the position of one of these
masers over a period of several years,
one will see an intriguing pattern.
Shown below are VERA measurements of a maser
in the Orion-KL region;
the figure shows the Right Ascension component of the
position as a function of time.

Now, if VERA can measure parallax with a precision of
0.03 mas, then it can reach far out into the Milky Way, too.
In fact, at the moment, VERA is one of our best tools
for measuring distances in the Milky Way,
beating Gaia quite handily.

Way back in the late 1990s, I discussed parallax and the
(at that time)
exciting new results from Hipparcos.
You might find some of the illustrations of the "raw-ish"
data from Hipparcos useful.

Of course, you will want to get your hands on the real Gaia
measurements. You have several choices:

Vizier catalog I/337
provides interactive searching and plotting
access to Gaia's TGAS catalog. I find TGAS more
useful than the other Gaia DR1 products,
since it includes parallaxes and proper motions.